Aircraft: MEDIUM TRANSPORT; LOW WING; 2 TURBOJET ENG
Phase of flight: Parked
Anomaly: Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural FAR; Deviation / Discrepancy - Procedural Published Material / Policy

Crew was departing on the final leg of [this trip]. This flight would take us from ZZZ to [our final destination]. Upon landing at ZZZ; crew discovered they were without cellular reception and the frequency solicited inflight inbound from Center for IFR communications outbound failed to reach any ATC Facilities. The PIC discovered that standing in the unattended county ops facility provided minimal cellular reception. A Center number was obtained through company resources and Center provided a TRACON number for IFR Clearance. A call to TRACON followed and a discussion of poor communications in the area ensued and an IFR Clearance was copied. I explained that crew would proceed to the aircraft and taxi but would be unable to call from any other location on the field and requested a window of opportunity (perhaps 10 mins) to depart. The controller (in words to the effect) explained we could; indeed; depart IFR and said something about taxiing out; whereupon; I explained we would be unable to call again.There was more to the discussion but I am unable to recall the content. This is where the confusion entered the pre-departure phase of flight. I interpreted the controller understood I was going to taxi out and would be unable to phone from anywhere else on the field and this was my pre-departure call and the controller believed I was going to phone again prior to departure (though I had explained this would not be possible). What I believe I heard in the moment were 'taxi out' and in my mind had interpreted as go ahead and equated it to 'proceed.' I am certain the tonality of my final words to the controller; 'oh; okay' contained an element of surprise as if being given a Clearance in an unusual fashion or circumstance. I justified the unusual Clearance to depart as local protocol as communications were an issue at ZZZ and the air park is a remote airport location. Just to verify one last time; I attempted cellular contact at the end of the runway without success. Following departure in VMC conditions (10+SM / CLR); we contacted TRACON and were asked if we departed on an IFR Clearance. (To our knowledge) we replied in the affirmative. TRACON advised us of a possible pilot deviation and provided a number to call.A discussion with TRACON followed arrival at [our destination] and we attempted to sort out the communication failures that had occurred at ZZZ. The final words on the subject couldn't be clearer; 'never depart without the magic words 'Cleared' followed by a void time.'The nature of the operation sometimes places crews in locations and situations that are; perhaps; non-standard and these non-standard 'procedures and protocols' occasionally begin to infiltrate more structured environments. In this case; because of the location and operations at the field; a less vigilant perspective developed and believing a less standard form of operations was at work reduced a crew member's situational awareness. Under the circumstances that led to this event (a remote unattended field; poor communications; casual chatting with ATC Clearance); a 'belief or understanding of what was expected' of flight crew and ATC was not shared. As mentioned in the narrative; 'never depart without the magic word 'cleared' followed by a void time.' Fractional Captain reported departing non-towered airport without IFR clearance.